GMM Estimation of Short Dynamic Panel Data Models With Error Cross Section Dependence

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1 GMM Estmaton of Short Dynamc Panel Data Models Wth Error Cross Secton Dependence Vasls Sara ds Ths verson: March 008 Abstract Ths paper consders the ssue of GMM estmaton of a short dynamc panel data model when the errors are correlated across ndvduals. We focus partcularly on the condtons requred n the cross-sectonal dmenson of the error process for the dynamc panel GMM estmator to reman consstent. To ths end, we demonstrate that cross secton ndependence (or uncorrelatedness) s not necessary rather, t su ces that, f there s such correlaton n the errors, ths s weak. We de ne a stochastc scalar sequence to be weakly correlated at any gven pont n tme f random varables su cently far apart n the sequence exhbt very lttle correlaton. Spatal dependence sats es ths condton but factor structure dependence does not. Consequently, the dynamc panel GMM estmator s consstent only n the rst case. Under weakly correlated errors, an addtonal set of moment condtons becomes relevant for each spec cally, nstruments wth respect to the ndvdual whch unt s correlated wth, denoted by j. We demonstrate that these extra moment condtons can be partcularly useful when the errors are subject to both weak and strong correlatons, a stuaton that s lkely to arse n practce. Smulated experments show that the resultng method of moments estmators largely outperform the conventonal ones n terms of both bas and RMSE. Key Words: dynamc panel data, spatal dependence, factor structure dependence, Generalsed Method of Moments. JEL Class caton: C3; C3; C33. Introducton In developng the theory of GMM estmaton of short dynamc panel data models, t s commonly assumed that the regresson errors are ndependently dstrbuted across ndvduals (see e.g. Anderson and Hsao, 98, pg. 598, Arellano and Bond, 99, pg. 78, Arellano, 993, pg. 88, Ahn and Schmdt, 995, pg. 7, Blundell and Bond, 998, page 8, and others). Ths assumpton s usually made for dent caton purposes Dscplne of Econometrcs and Busness Statstcs, Unversty of Sydney, SW 006, Australa. Tel: ; e-mal: v.sara ds@econ.usyd.edu.au.

2 rather than descrptve accuracy wth the hope, presumably, that by condtonng on a su cent number of explanatory varables, what s left over can be treated as a purely dosyncratc dsturbance that s uncorrelated across ndvduals. On the other hand, n emprcal applcatons of GMM estmaton ths rather strong assumpton s somewhat relaxed by allowng for common varatons n the dependent varable at any gven pont n tme usng two-way error components dsturbances (e.g. Arellano and Bond, 99, pg. 88, Blundell and Bond, 998, pg. 37, Bover and Watson, 005, pg. 975). In practce, however, a + f t + t formulaton s unlkely to be adequate to remove all correlated behavour n the errors and ths may result n msleadng nferences and even nconsstent GMM estmators (Sara ds and Robertson, 007). Error cross secton dependence may arse for varous reasons n practce; for example, t may be due to the presence of spatal correlatons spec ed on the bass of economc and socal dstance (Conley, 999) or relatve locaton (Anseln, 988), as well as due to the presence of unobserved components that gve rse to a common factor spec caton n the dsturbances wth a xed number of factors (e.g. Goldberger, 97, and J oreskog :: and Goldberger, 975). Methods that account for a mult-factor error structure have been proposed by Robertson and Symons (000), Coakley, Fuertes and Smth (00), Phllps and Sul (003), Moon and Perron (004), Ba (005), Pesaran (006) and others. However, these methods are theoretcally just ed n panels where the number of tme seres observatons (T ) s large. To the best of our knowledge, no study exsts that accounts for spatal correlatons n a short dynamc panel data model. The present paper deals spec cally wth the ssue of GMM estmaton of a short dynamc panel data model when the errors are not ndependent across ndvduals. A major focus les on the condtons requred n the cross-sectonal dmenson of the error process for the dynamc panel GMM estmator to reman consstent. To ths end, we demonstrate that ndependence, or uncorrelatedness, s not necessary for GMM consstency or asymptotc e cency rather, t s su cent that, f there s such correlaton n the errors, ths s weak. We de ne a stochastc scalar sequence to be weakly correlated at any gven pont n tme f random varables su cently far apart n the sequence exhbt very lttle correlaton. Therefore, a weakly correlated sequence s asymptotcally uncorrelated. Conversely, a sequence s strongly correlated f random varables reman correlated no matter how far apart the le n the sequence. We show that the spatal approach to modellng error cross secton dependence, whch typcally assumes unform boundedness of the row and column sums of the weghtng matrx, sats es asymptotc uncorrelatedness, although t s more restrctve n the sense that the latter does not requre unform boundedness. On the other hand, under factor structure dependence the errors are strongly correlated and therefore the GMM estmator s not consstent. The two-way error components model volates asymptotc uncorrelatedness too, albet the problem can be dealt n ths case va tme-demeanng of the observatons. However, careful analyss needs to be made n ths case because the aforementoned transformaton In an n uental paper, Phllps and Sul (007) analyse the mpact of error cross secton dependence on the dynamc Fxed E ects (FE) estmator. Assumng that the unobserved tme-spec c ndvdual-nvarant e ect s treated as stochastc.

3 nduces some dependency among the ndvdual equatons and therefore the moment condtons are not vald anymore for nte, a result that s usually gnored n the lterature. In addton, ths paper shows that when the errors are weakly correlated n the way de ned above, then for each ndvdual there s an addtonal set of moment condtons that becomes relevant n partcular, nstruments wth respect to the ndvdual whch unt s correlated wth, denoted by j. We demonstrate that these extra moment condtons can be partcularly useful when the errors are subject to both weak and strong correlatons, a stuaton that s lkely to arse n practce. Pesaran and Tosett (007) consder ths stuaton as well, for a model wth no lags of the dependent varable on the rght-hand sde and T su cently large. The structure of the paper s as follows. The followng secton spec es the panel regresson model n a way that encompasses common factors and spatal dependence. Secton 3 revews the standard moment condtons used n GMM estmaton under twoway error components dsturbances. Secton 4 addresses the ssue of consstency for the dynamc panel GMM estmator when the ndependence assumpton across ndvduals s relaxed. Secton 5 shows that under weakly correlated errors, addtonal moment condtons become relevant for each ndvdual, whch arse from the ndvdual(s) whch unt s correlated wth. Secton 6 demonstrates the valdty of these extra moment condtons under both weakly and strongly correlated errors and the followng secton analyses the propertes of the resultng GMM estmators, ncludng cases where the problem of weak nstruments apples. The performance of these estmators s nvestgated n Secton 8 usng smulated data. A nal secton concludes. Model Spec caton We focus on dynamc panel data models of the followng rst-order autoregressve form: y ;t y ;t + ;t ; ; :::; and t ; :::; T ;t + u ;t u ;t MX X m w;j m m j;t + ;t 0 dag W 0 t M + ;t () m j where y ;t s the dependent varable of ndvdual at tme t, s a xed parameter to be estmated wth jj <, and t s a composte error term that conssts of an ndvdualspec c tme-nvarant unobserved e ect and a weghted sum of purely dosyncratc components, where ; :::; M 0 s an M vector, W 6 4 w; w; w; w; w; w; : : : : w; M w; M w; M ; M, t ;t ;t ;t ;t ;t ;t : : : : M ;t M ;t M ;t ; M, ()

4 and M s a M column vector of ones. We make the followng assumptons: Assumpton : d 0;. Assumpton : m t d 0; and m t d 0;. Assumpton 3: E (y ; ;t ) 0; for ; :::; and t ; 3; :::; T. h Assumpton 4: m s non-stochastc and bounded wth lm P! 6 h 0, lm P! 0 j for j and 0 otherwse, where, s a dagonal postve sem-de nte matrx and s a M vector such that jj jj < B <. Assumptons -3 are standard n the GMM lterature. Assumpton can be easly relaxed by allowng t MA(k), where k s a small postve nteger. Assumpton 3 ensures that su cently lagged values of y t wll be uncorrelated wth the rst-d erence of t and thus they wll be avalable as nstruments. Assumpton 4 s equvalent to requrng that has nte mean and varance and t s uncorrelated across for all m f were stochastc, whch would be sats ed under say d ( ; ). ote that all the results dscussed below extend n an obvous fashon to hgher order autoregressve processes as well as to panel autoregressve dstrbuted lag models. Model () can be wrtten n a more compact form as follows: y y + + u, u MX m (W m I T ) m + (3) m where y (y ; :::; y ) 0 s a (T ) matrx wth y (y ; ; :::; y ;T ) 0, y (y ; ; :::; y ; ) 0 s a (T ) matrx wth y ; (y ; ; :::; y ;T ) 0, ( ; :::; ) 0 wth T beng a (T ) column vector of ones, m dag ( m ; :::; m )0 T s a (T ) vector, W m s a weghtng matrx, m ( m ; :::; m )0 wth m m ;; :::; m 0 ;T and ( ; :::; ) 0 wth ( ; :::; ;T ) 0. The composte error term, u, has a exble structure n that t can characterse varous forms of cross secton dependence, whch nclude dependence that s due to the presence of unobserved common factors as well as spatal correlatons n the error term, dependng on the structure of W m. Spec cally, the mult-factor structure arses from (3) by settng all elements of W m, denoted by wm ;j, equal to such that u w m ;j for ; :::;, m ; :::; M (4) MX m f m +, where f m (f m ; :::; ft m ) 0 X and ft m m ;t. (5) m 4

5 In ths case we have E (ft m ) 0, var (ft m ) m and cov ft m ; ft m 0 for > 0 and all m. Therefore, E (u ;t ) 0; var (u ;t ) P M m (m ) m + and ( PM cov (u ;t ; u +k;s ) m m m +k m for t s 0 otherwse (6) The Spatal Movng Average (SMA) process arses from (3) by settng M, wth jj <, and W W equal to a sparse matrx populated prmarly wth zeros. For nstance n a crcular 3 SMA() process, u equals u (W I T ) + (7) wth W gven by W : : : : : : : : : : : : 0 : : : : : : : : : 0 : : : : : : : : : In ths case we have E (u ;t ) 0, var (u ;t ) + and cov (u ;t ; u +k;s ) for (mod ) + and t s 0 otherwse (8) (9) where (mod ) s the modulo operator, de ned as the remander after numercal dvson of by to obtan nteger values. Thus, for ; :::;, (mod ) + + and for, (mod ) +. SMA processes of hgher order can be accomodated straghtforwardly. Assumng nvertblty, the Spatal Autoregressve (SAR) form can be obtaned usng an n nte SMA representaton 4. The Spatal Error Components (SEC) form arses n a way smlar to a SMA form wth the only d erence beng that 6 n (7) whle (8) ncludes non-zero values on the man dagonal. In ths case we have E (u ;t ) 0, var (u ;t ) + and cov (u ;t ; u +k;s ) for (mod ) + and t s 0 otherwse (0) Fnally, t follows that by mposng approprate restrctons on m, w;j m and m ;t, (3) can easly accomodate mxture cases too, where both spatal correlatons and common unobserved factors are present n the errors. We consder estmaton of ths type of models n Secton 6. 3 See e.g. Baltag, Bresson and Protte (007). 4 In ths case, W s not sparse, however ts elements wll declne wth a dstance measure that ncreases su cently rapdly as the sample ncreases. For nstance, Stetzer (98) models the dstance decay by a negatve exponental functon, w j exp ( d j), 0 < <, wth d j denotng the dstance between ndvduals and j. 5

6 3 Moment Condtons n Standard GMM Estmaton Typcal GMM estmaton of lnear dynamc panel data models of the form gven n () mposes m 0 for all and m, such that any form of dependence n the error process across ndvduals, whether ths s spatal or subject to a factor structure, s ruled out 5. Consequently, applyng rst-d erences n () yelds y ;t y ;t + ;t ; ; :::; and t 3; :::; T () Under Assumptons -3 the followng (T ) (T ) moment condtons become avalable E (y ;t s ;t ) E (y ;t s ;t ) 0; for t 3; :::; T and s ; :::; t. () On the other hand, n emprcal applcatons t s common practce to generalse the error structure by allowng for common varatons n the dependent varable usng a two-way error components formulaton 6 : ;t + f t + ;t. (3) If f t s treated as non-stochastc, the moment condtons gven n () are not vald anymore because the expectaton of f t s equal to f t tself. Hence! E (y ;t s ;t ) E + X X ;t s + f t s (f t + ;t ) X f t s f t 6 0. (4) If, nstead, f t s treated as stochastc and serally uncorrelated, the moment condtons gven above reman vald because E [ P f t s f t ] 0. However, the sample counterpart of (4) does not converge to ts expectaton for nte T for reasons that wll become clear n the next secton. Instead, X [y ;t s ;t ] X p f t s f t! 0 (5) Transformng the observatons n terms of devatons from tme-spec c averages elmnates both of these problems by removng the common tme e ect from the regresson error: t t t ( ) + (f t f t ) + ( t t ) + t. (6) However, the cross-sectonally demeaned transformaton nduces some dependency among the ndvdual equatons and therefore the moment condtons on the trans- formed observatons are not vald for nte,.e. E y ;t s ;t E y ;t s ;t 6 0, 5 See Secton for related references. 6 Vz. footnote 5. 6

7 where y ;t s y ;t s y t s, and smlarly for the remanng varables. As grows large, ths dependency dsappears; n partcular, de nng y o ;t s y ;t s e yt s and o t ;t e t, where eyt s P f t s and e t f t, we have p p p X hy ;t s ;t X y;t s e yt s X h y o ;t s o ;t y t s e yt s ;t e t t e t + o p () (7) snce t e t Op, y t s e yt s O p, P yo ;t s O p () and P o t O p (). Therefore, gven that y o ;t s o ;t are ndependent across, a sutable CLT (Central Lmt Theorem) ensures that p X h y o ;t s o ;t d! 0; V ar p y o ;t s o ;t (8) Hence, de nng y ; y Z ; y ; y ; y ; y ;T ; 6 4 ;3 ;4. ;T 3 7 5, (9) and Z (Z ; Z ; :::; Z ) 0, the rst-d erenced GMM estmator equals b DIF GMM y0 Z A b Z 0 y y0 Z A b Z 0 y (0) wth y (y ; :::; y ) 0, y (y ;3 ; :::; y ;T ) 0, y ; (y ; ; :::; y ; ) 0 and y ; (y ; ; :::; y ;T ) 0. b A s some weghtng matrx that sats es ba A p! 0 () where A s a non-stochastc sequence of postve de nte matrces. Alternatve choces of A b lead to d erent GMM estmators, whch are all consstent but they d er n terms of e cency. The asymptotcally e cent DIF GMM estmator sets A b equal to the nverse of the covarance matrx of the moment condtons 7 that s, A b 7 See Hansen (98). 7

8 p Est:Asy:V ar Z0, assumng that ths matrx exsts and s nte postve defnte. When t s homoscedastc A b can be approxmated by Z0 HZ, where and H 6 4 H I H () : : 0 0 Hence t s clear that the weghtng matrx s a functon of the Kronecker product between two dstnct matrces, the former of whch re ects cross secton dependence n the error structure (whch, for large, s zero n the present case and hence the use of the dentty matrx) whle the latter re ects tme seres dependence n the error structure, and n partcular rst-order seral correlaton, whch s nduced by rst-d erencng the observatons. ote that snce the ndvdual equatons are ndependent across for large, A b can also be wrtten as A b P Z0 H Z and therefore an equvalent expresson for (0) s gven by b DIF GMM X y 0 ; Z! ba X! Z 0 y ; X y 0 ; Z! ba X (3) Z 0 y! When the ndvdual observatons are not ndependent across, (4) s not equvalent to (0). The standard rst-d erenced GMM (DIF GMM) estmator may have poor nte sample propertes n terms of bas and precson when! or!. As a result, Blundell and Bond (998) developed an approach outlned n Arellano and Bover (995), whch combnes the equatons n rst-d erences wth the equatons n levels, usng y ;t as an nstrument for the lagged dependent varable, y ;t : E (y ;t ;t ) 0; for t 3; 4; :::; T (5) Ths approach gves rse to a system GMM (SYS GMM) estmator, whch s vald provded that the devatons of the ntal observatons from the long-run convergent values are uncorrelated wth the ndvdual-spec c, tme-nvarant e ects that s, E y ; 0. (6) If common tme e ects are ncluded n the error process, what s requred s that X plm! y ; 0. (7) 8 (4)

9 Thus, de nng Z sys 6 4 Z y ; 0 : :... : ; sys, 0 0 y ;T and Z sys Z sys ; Z sys ; :::; Z sys 0, the SYS GMM estmator s gven by b SY S GMM Y0 Z sys A;sys b Z sys0 Y Y0 Z sys A;sys b Z sys0 Y (8) where Y (Y ; Y ; :::; Y ) 0 0,, Y y ;3 ; :::; y ;T ; y ;3 ; :::; y ;T, Y Y ; ; Y ; ; :::; Y ; and Y ; y ; ; :::; y ;T ; y ; ; :::; y ;T. Wth homoscedastc errors the optmal choce of b A ;sys s gven by 8 where H sys equals ba ;sys Zsys0 H sys Z sys (9) H sys H C C 0 I (T ) wth I (T ) I I T and C I C, where C takes the value of on the man dagonal, on the rst lower o -dagonal and zero otherwse. The next secton addresses the ssue of consstency for the dynamc panel GMM estmator when the restrcton m 0 8, m s relaxed. 4 The Consstency of the Dynamc Panel GMM Estmator under Error Cross Secton Dependence When m s bounded and d erent from zero n (), the structure of W m wll be crtcal upon the asymptotc propertes of the GMM estmator. Wthout loss of generalty, we wll mpose M for the remanng P of ths secton, such that the error process becomes equal to ;t + j w ;j j;t + ;t wth j j < B <, E ( ;t ) 0 and E ;t < B <. We rstly de ne the concept of a weakly correlated process. Let t, be the scalar sequence ( ;t; ;t ; 3;t ; :::). There are (T ) such scalar sequences, for t ; :::; T. 8 For 0; see Wndmejer (000) and Kvet (007). (30) 9

10 De nton The scalar sequence n t, o exst non-negatve constants t;s ; 0 s sad to be weakly correlated f there, where 0 t;s and t;s E ( ;t ; +;s ) E ;t E +;s for all 0, (3) such that for all t and s. X t;s < (3) otce that t;s s merely an upper bound for the correlaton between ;t and +;s, assumng that E ( ;t ) 0 for and all t, s. Snce t s only postve correlaton that matters, f ;t and +;s are negatvely correlated, we can set t;s 0. Thus, De nton mples that random varables su cently far apart n the sequence exhbt very lttle correlaton. Remark Observe that for P t;s <, t s necessary that t;s o () and t s su cent that t;s o. Therefore, a weakly correlated process s asymptotcally uncorrelated. Conversely, a sequence that s not asymptotcally uncorrelated that s, where random varables reman correlated no matter how far apart they le n the sequence, s sad to be strongly correlated. Theorem 3 Let t, be the scalar sequence ( ;t; ;t ; 3;t ; :::), where ;t + P j w ;j j;t + ;t, wth j j < B <, E ;t < B < and jjwjj max X j jw ;j j o (33) Then t s weakly correlated, or asymptotcally uncorrelated. Proof. See Appendx A. ote that condton (33) n Theorem (3) s more general than a unform boundedness condton for the row and column sums of W (typcally employed n spatal models), whch s stated as follows 9 : X jw ;j j B w < 8 j and X jw ;j j B w < 8 (34) j Ths s because unform boundedness s subject to (33) but not vce versa. For nstance, we may have jw ;j j 3 8, j, n whch case the row and column sums of W are not bounded because P jw ;jj 3 and thefore t s growng wth. 9 See e.g. Kapoor, Kelejan and Prucha (007, pg. 06) and Lee (007, pg. 49). 0

11 However, condton (33) s stll sats ed. As a result, any spatally correlated process that sats es (34) s weakly correlated, or asymptotcally uncorrelated. On the other hand, the factor structure sets jw ;j j 8, j and so t volates (33). Hence t provdes an example of a process that s not weakly correlated. As a matter of fact, when jw ;j j there are M unobserved varables, ft m ( P m t + ::: + m t ), whch are common for all and therefore ther e ect does not dmnsh no matter how far n the sequence two random varables, ;t and +;t, are. As a result, the factor structure dependence s an example of a strongly correlated process. The two-way error components model wth stochastc f t s a restrcted case of the sngle-factor structure because t sets for all although t retans the same form for w ;j. Therefore, t provdes another example of a strongly correlated process, albet the correlaton can be removed n ths case for large by transformng the data n terms of devatons from tme-spec c averages. otce that condton (33) does not mply that the t, sequence s spatally ergodc because the row sums of W need not necessarly be the same, n whch case the elements of the sequence are not dentcally dstrbuted. Furthermore, condton (33) does not requre that the sequence s a mxng process ether n the sense that the elements of the sequence can be asymptotcally uncorrelated but not asymptotcally ndependent 0. Remark 4 Pesaran and Tosett (007) de ne the scalar sequence z t, to be weakly dependent at any gven t f ts (weghted) average converges to ts expectaton n quadratc mean. Spec cally, let w t denote a weght that sats es P w t O h P and w t w t O for any, and let I t be the nformaton set at tme t contanng at least z t, z t ; ::: and w t, w t ; :::, where z t z t ; :::; z t 0 and w t w t ; :::; w t 0. Then the sequence z t, s weakly dependent f! lm var X w t z t! I t 0. Under ths de nton, the followng factor structure process u ;t f t + ;t, where s non-stochastc and bounded f t ::d 0; f, ;t ::d 0;, (35) P s weakly correlated so long as lm! 0. Ths s not the case, however, usng De nton snce t s straghtforward to show that t;t 9 0 as! and therefore P t;s s not bounded. Intutvely, no matter how large s, u ;t and u +;t reman correlated n a non-neglgble way regardless of whether! 0 or not. 0 Of course, ths requres a strengthenng of the moment restrctons namely, E ju j < B u <, as opposed to say E ju j c < B u < for c > ). See Pesaran and Tosett (007), Theorem 6, page 5.

12 The followng theorem (due to Whte, 00, Theorem 3.57) provdes a law of large numbers for weakly correlated sequences. Theorem 5 Let t, ( ;t; ;t ; 3;t ; :::) be a scalar sequence wth weakly correlated elements, such that E ( ;t ) 0 and E ;t < B <. Then X ;t E ( ;t )! p 0 (36) Proof. It follows drectly from Stout (974, Corollary.4.) and the Kronecker lemma. Theorem 3 shows that so long as (33) holds true, ;t s weakly correlated, or asymptotcally uncorrelated across. In turn, accordng to Theorem 5, the latter mples that the rst sample centered moment of t wll converge n probablty to zero. The followng corollary provdes the extra condton necessary to valdate the moment condtons gven n () under weakly correlated errors: Corollary 6 Let t, and u t, be two scalar sequences ( ;t; ;t ; 3;t ; :::) and (u ;t ; u ;t ; u 3;t ; :::) that satsfy jjwjj o ndvdually and are, therefore, weakly correlated. The product of these sequences wll also satsfy jjwjj o and wll be weakly correlated. As a result, we have plm! ts+ ts+ X ts+ plm! X X plm! (y ;t s ;t ) + X u ;t s! u ;t X u ;t s u ;t 0 for s ; :::; T. (37) where the last lne holds true because both f ;t s ; g and f ;t, g fu ;t, g, are weakly correlated and so ther product s also weakly correlated or asymptotcally uncorrelated across, wth expected value equal to zero. Thus, the moment condtons used by DIF GMM reman vald. Remark 7 Observe that when a weakly correlated process s de ned as n Remark 4, the sample average over of the product between f ;t s, g and f ;t, g does not necessarly converge to ts expectaton; for nstance, for the sngle-factor process gven n (35) and assumng that lm P! 0, we have P u ;t s E (u ;t s )! 0 and P u ;t E (u ;t )! 0. However, the sample average P u ;t su ;t converges to f t sf t, where plm! P, despte the fact that E (u ;t s u ;t ) 0 for s ; :::; t.

13 Snce asymptotc uncorrelatedness encompasses spatal dependence, t follows that DIF GMM s consstent under spatally correlated errors. On the other hand, under factor structure dependence the correlaton between t and jt perssts no matter how far apart ndvduals and j are. Therefore the law of large numbers provded above P breaks down and plm! y ;t s ;t 6 0, despte the fact that E (y ;t s ;t ) 0. SYS GMM also remans consstent under weakly or spatal correlated errors because! X X X plm! (y ;t ;t ) plm! u ;t t 0. t3 t3 (38) In summary, t has been shown that the dynamc panel GMM estmator does not requre cross-sectonally ndependent errors for consstency rather, t su ces that, f there s such dependence, ths s weak n the way de ned above at any gven pont n tme. Theorem 3 shows that ths holds true under condton (33), whch s more general than unform boundedness of the row and column sums of W. The factor structure n the error process volates ths condton and therefore the standard GMM estmator s not consstent n ths case. 5 Addtonal Moment Condtons Under Spatal Dependence Suppose that the errors are spatally correlated but satsfy condton (33). It turns out that not only DIF GMM and SYS GMM are consstent, but also that there s an addtonal set of moment condtons whch becomes relevant n ths case. In partcular, we consder the basc model gven n (3) and for smplctly we mpose a SMA() error process that s, M, wth jj <, and W s gven by (8). Hence, the model becomes equal to y ;t y ;t + ( + j;t + ;t ) ; ; :::; and t ; :::; T (39) where j (mod ) + 3. proposton demonstrates: In ths case, an nterestng result arses, as the followng Proposton 8 Under Assumptons -3, the panel autoregressve model n (39) mples that for each ndvdual there s an addtonal set of moment condtons that becomes relevant wth respect to a d erent cross secton, ndvdual j, both n the rst-d erenced equatons and those n levels. In partcular, we have Moment Condtons for DIF GMM: X plm! y j;t s ;t 0; for s ; :::; T, (40) ts+ See also Sara ds and Robertson (007). 3 See also (9). SMA processes of hgher order can be accomodated n a smlar fashon. 3

14 wth X plm! ts+ y j;t s y ;t (T s) + (4) Moment Condtons for SYS GMM (assumng that (6) s also sats ed): X plm! y j;t ;t 0 (4) wth X plm! Proof. See Appendx B. t3 y j;t y ;t t3 (T ) + (43) ote that the error term of the regresson model does not necessarly need to be of a spatal MA form. In fact, t s straghtforward to show that these moment condtons are relevant under SAR and SEC errors or under more general spatal processes. Most notably, these moment condtons reman vald under both weakly and strongly correlated errors, as t wll be shown n Proposton (9). In the case of (39), we have E 0 E ( (W I T ) +) ( (W I T ) +) 0 E [ (W I T ) + I T ] 0 W 0 I T + IT [ (W I T ) + I T ] (I H ) W 0 I T + IT [ (W I T ) (I H ) + I T (I H )] W 0 I T + IT (W H ) W 0 I T + (W H ) + (I H ) W 0 I T + (I H ) W W 0 + W + W 0 + I H (44) and therefore W W 0 + (W + W 0 ) + I replaces I n the expresson for the weghtng matrx of DIF GMM n (). A smlar pont apples to I n (30) for SYS GMM. Of course, n practce s unknown; one opton s to replace wth an arbtrary value (say 0:5) at rst stage, and then obtan an estmate of by solvng the followng quadratc equaton: b t r t () b t + r t () 0 (45) where r t () Est:Correlaton (b ;t ; b j;t ) and b ;t s the rst-stage resdual of unt for t ; :::; T. (45) has two solutons for each t, but gven that rt () b t + b t one root s the recprocal of the other, whch mples that the estmator for at tme t equals b t p 4r t () r t () The other soluton can be ruled out snce t wll have an absolute value greater than one, whch s not possble gven the restrcton jj <. A smple average b T P bt can then be constructed to provde an estmate of b. 4 (46)

15 6 Consstent GMM Estmaton under both Spatal and Factor Error Structure The moment condtons analysed n the prevous secton can be partcularly useful n general error processes that nclude unobserved common factors as well as omtted varables that are spatally correlated. Ths s because whle the standard moment condtons n () and (5) are nvaldated n ths case 4, t turns out that the moment condtons obtaned from a d erent cross secton, ndvdual j, are stll vald. In partcular, consder agan the error process of the basc model gven n (3) and suppose that whle W m 0 for some m (e.g. m ; :::; M), there s at least a sngle W that sats es condton (33) of Theorem 3. Let ths be denoted by W M+ and be equal to (8), although t should be clear by now that any W that sats es condton (33) or unform boundedness wll do. In ths case, the basc model can be rewrtten as y ;t y ;t + ;t, ;t a + u ;t, u ;t MX m ft m + ;t + j;t (47) wth ft m ( P m t + ::: + m t ), j (mod ) + and jj <. A smlar error process that s subject to both spatal correlatons and common unobserved factors s studed by Pesaran and Tosett (007). Expressng (47) n terms of devatons from tme-spec c averages and usng rstd erences yelds y ;t y ;t + ;t ; ;t m MX m ft m + ;t + j;t (48) m The followng proposton demonstrates an mportant result: Proposton 9 Under Assumptons -4, the panel autoregressve model n (48) can be estmated consstently usng method of moments estmators that rely on the followng moment condtons wth Moment Condtons for DIF GMM: X plm! y j;t s ;t 0; for s ; :::; T, (49) ts+ X plm! ts+ y j;t s y ;t (T s) + (50) 4 The use of other nstruments wth respect to ndvdual wll not help ether, unless these nstruments are not functons of (lagged values of) y and certan regularty condtons hold true, such as those n Sara ds, Yamagata and Robertson (007). 5

16 Moment Condtons for SYS GMM: plm! wth X plm! Proof. See Appendx C. X t3 y j;t ;t 0 (5) y j;t y ;t t3 (T ) + (5) The above mples that the model gven n (47) can be estmated consstently usng a smple IV estmator that employs y j;t as an nstrument for y ;t, or a rst-d erenced GMM estmator that nstruments y ;t by y j;t s for s ; 3; :::; and a system GMM estmator that uses y j;t as an nstrument for y ;t n the levels equatons. Ths s because the correlaton between y j;t s and y ;t (or between y j;t and y ;t n levels) s non-zero whle the correlaton between y j;t s and ;t (and y j;t and ;t n levels) remans zero. Therefore, de nng Z MM Z MM ; :::; Z MM 0 wth Z MM 0 y j; ; y j; ; :::; y j;t as well as the followng matrces of nstruments 3 y j; Z y 0 y j; y j; ; y j; y j; y j;t ;3 ;4. ;T 3 7 5, (53) and Z ysys 6 4 Z y y j; 0 : :... : ; sys, (54) 0 0 y j;t Proposton 9 mples that the followng moment estmators are vald: b y IV Z 0 MMy Z 0 MM y, (55) and b y DIF GMM b y SY S GMM y0 Z ba y Z 0 y y0 Z ba y Z 0 y (56) Y0 Z sys A b y ;sys Z sys0 Y Y0 Z sys A b y ;sys Z sys0 Y. 6 (57)

17 ba y s the weghtng matrx of the two-step rst-d erenced GMM estmator, whch can be estmated from ba y Z0 bb 0 H Z (58) where H has been de ned n (3) and b s an (T ) matrx of resduals, obtaned from a rst-step rst-d erenced GMM estmator. otce that the least-squares estmate of bb 0 n (58) s rank de cent because t s an matrx and has rank T. The matrx nsde the square brackets of (58) s also rank de cent because t s a square matrx of order (T ) and has rank (T ) : However, A b y s a square matrx, whch has rank equal to mn ; (T ). Therefore, provded that we do not use too many nstruments,.e. (T ) ; (58) wll be of full rank and the weghtng matrx wllexst. ba y ;sys s the weghtng matrx of the two-step system GMM estmator, whch can be estmated from ba y ;sys Zsys0 Q b Z sys (59) wth b Q beng equal to bq bb 0 (H ) 0 0 bb 0 (I T ) (60) 7 Propertes of GMM Estmators To nvestgate the propertes of these moment estmators we follow the approach by Blundell and Bond (998) and we consder the case where T 3, for whch there s only a sngle nstrument avalable for the endogenous regressor, both n the rst-d erenced equatons and those n levels. In ths way, DIF GMM and SYS GMM reduce to smple nstrumental varable estmators and the correspondng rst-stage regressons may help to analyse the strength of the nstruments used as a functon of the parameters of nterest n more general cases. 7. Equatons n Frst-D erences For the equatons n rst-d erences, the rst-stage regresson s gven by y ; d y j; + w (6) where w s an error term. The ordnary least-squares estmate of d, whch re ects the strength of the relaton between the nstrument and the endogenous regressor, s equal to P b d y j; y ; (6) P y j; 7

18 Usng Assumptons -4 n model (47) t s straghtforward to show that the plm of b d equals plm! b d P ( ) plm! y j; y ; + plm! P yj; y ; y ; ( ) ( ) plm! P y j; ( ) w 0 w [w 0 ( ) w ] where + and w P s s f s. Thus, we can see that for xed T the plm of b d depends on varous parameters, namely ; ; ; and : For example, as! the plm of the estmator converges to zero, whch mples that the correlaton between y j;t and y ;t becomes weak. The ntuton behnd ths s llustrated n the followng gure, whch shows two cases of when the values of ;, and are held xed: (63) Weak nstruments wth cross secton dependence. When 9, y j; s correlated wth y ; and snce Cov y j; ; ;3 0 y j; s a vald nstrument. ote that the use of y ; as an nstrument s not vald here because 6 0 and therefore Cov y ; ; ; On the other hand, as! the correlaton between y j; and y j; becomes weak; ths s because the lnk between y j; and y ; s not e ectve anymore snce y j; s poorly correlated wth y j; ; whle the lnk between y j; and y ; does not help ether because y ; s poorly correlated wth y ;. When there s no varaton n the factor loadngs across, 0 and the plm of b d remans non-zero but of course n ths case y ; s also vald as nstrument. On the other hand, for a gven non-zero value of and jj < the plm of the estmator converges to zero as ether! or!. The former result s smlar to Blundell 8

19 and Bond (998). Interestngly, the same appears to apply for the rato between and. Intutvely, ths s because the contrbuton of the spatal component of the error process n ;3 (and thereby the correlaton between y and y j ) dmnshes wth hgh values of and ncreases wth hgh values of. 7. Equatons n Levels For the equatons n levels the rst-stage regresson s gven by and the least squares estmator of l equals y ; l y j; + w l (64) b l P y j; y ; P y j; (65) Usng Assumptons -4, t s straghtforward to show that the plm of b l equals plm! b l plm! P y j; y ; P plm! y j; y ; P plm! y j; + + ( + ) [w 0 ( ) w ] plm! P y j; ; where w P f. Here we can see that as! the above expresson converges to plm! b d (66) + + w 0 ( ) w (67) and so y j; remans nformatve as an nstrument for y ;, provded of course that 6 0. In addton, when 0 the random element n (67) dsappears and the expresson becomes equal to a constant number spec cally, plm! b l +. Smlarly to (63) ; the plm of b l converges to zero as! for the same reason that has been dscussed prevously that s, because the contrbuton of the spatal component of the error process n 3 dmnshes. 8 Small Sample Propertes of Moment Estmators Ths secton nvestgates the nte-sample performance of the varous estmators proposed n ths paper usng smulated data. The man focus of the analyss les on the mpact of the relatve mportance of the unobserved factors n the total error process for d erent values of, T and. 9

20 8. Monte Carlo Desgn The underlyng data generatng process s gven by y t y t + + u t ; u t f t + t + jt, ; ; :::; ; t 48; 47; :::; T. (68) where d 0; ; t d 0;, ft d 0; f and j (mod ) : Also, the factor loadngs are drawn from du [ 0:5; 0:5] (69) The performance of GMM estmaton depends crucally upon the rato of the two varance components, a and u t ; on var(y t ) as shown n (63). Ths mples that as the value of ncreases, or the amount of cross-sectonal dependence decreases, the mpact of on var (y t ) wll tend to become larger and thereby comparsons across experments wth d erent levels of cross secton dependence wll not be vald. To control ths rato we use the followng smple result! var (y t ) var + X s u t s ( ) + u (70) s and we set [( ) ( + )] u wth 5. In addton to u, the performance of the estmators wll depend on the proporton of u attrbuted to the factor structure n u t hereafter ths proporton s denoted by (d), d ; :::; 4. Therefore, notcng that u f + f + + (7) and normalsng f, we can produce the followng result (d) (d) + + (7) Snce the values of and are determned solely by (69) and so they are xed, normalsng 0:5 mples that wll change only accordng to (d). As ths rato ncreases, the mpact of the factor structure n the error process wll rse. We choose the followng values for (d) : 8 >< >: Low mpact of factor structure on u t : () 3 Medum mpact of factor structure on u t : () Medum-to-hgh mpact of factor structure on u t : (3) 3 Hgh mpact of factor structure on u t : (4) 34 5 See Kvet (995) and Bun and Kvet (006). 0

21 We consder 400; 800 and T 6, 0, snce our focus s T xed,!. alternates between 0:5; 0:7 and 0:9: The ntal value of y t has been set equal to zero but the rst 50 observatons have been dscarded before choosng the sample, so as to ensure that the ntal zero values do not have an mpact on the results. All experments are based on,000 replcatons. 8. Results Tables - report the smulaton results n terms of the mean value of b r, where r denotes the r th replcaton, and RMSE for each of the estmators used n the experment. FE s the xed e ects estmator, IV s the smple nstrumental varables estmator that uses y t as an nstrument for y t and DIF and SYS denote the rst-d erenced and system GMM estmators respectvely 6. The superscrpt ndcates that the correspondng estmator uses nstrument(s) wth respect to another cross secton, unt j. As expected, the performance of all estmators depends on (d), the value of and the sze of T and. Spec cally, as the value of ncreases for a gven value of, T and, the estmators su er a rse n bas and n RMSE. Ths s natural because as the relatve mpact of the factor structure n the total error process ncreases, the nvaldty of the nstruments used wth respect to unt tself (such as n IV, DIF and SYS) s magn ed. For the estmators that make use of nstruments wth respect to unt j, the rse n bas and RMSE s also ntutve because as ncreases, the contrbuton of the spatal component n the error process and thereby the correlaton between y t and y jt dmnshes. Havng sad that, two thngs are clear from these results; rst, IV, DIF and SYS outperform IV, DIF and SYS respectvely under all crcumstances. Second, the relatve performance of IV, DIF and SYS mproves wth larger values of. Ths s also ntutve ultmately, as! 0 the factor structure n the error process dmnshes and the asymptotc bas of IV, DIF and SYS approaches zero. otce also that n terms of RMSE, SYS performs better than DIF, whch performs better than IV, wth the relatve d erence n performance beng ncreased accordng to the value of. As T rses, the performance of the estmators mproves wthout excepton. Fnally, t s mportant to emphasse that as the sze of ncreases, the bas and RMSE of IV, DIF and SYS decreases consderably. Ths s not the case for the conventonal estmators, IV, DIF and SYS, the performance of whch f anythng deterorates wth larger values of. 9 Concludng Remarks Error cross secton dependence s an ncreasngly popular research topc n the analyss of panel data. Despte ths fact, the ssue has mot attracted much attenton n GMM 6 DIF and SYS are estmated n two steps and they use y t and y t 3 as nstruments for y t n the rst-d erenced equatons. Furthermore, SYS GMM uses the optmal weghtng matrx (when 0), as derved n Wndmejer (000).

22 estmaton of short dynamc panels, where t s commonly assumed that the regresson errors are are ndependent across. Ths paper has shown that, n fact, ndependence or uncorrelatedness s not necessary for GMM consstency or asymptotc e cency rather, t s su cent that, f there s such correlaton n the errors, ths s weak n the sense that any two errors that le su cently far apart n the stochastc sequence exhbt very lttle correlaton at any gven pont n tme. If ths condton s not sats ed, the errors are sad to be strongly correlated. Spatal dependence presents an example of weakly correlated errors whle the factor structure dependence provdes an example of strongly correlated errors. Therefore, the standard dynamc panel GMM estmators that exst n the lterature reman consstent under spatally correlated errors but not so under a factor structure. When the errors are weakly correlated there are addtonal moment condtons that arse n partcular, nstruments wth respect to the ndvdual(s) whch unt s correlated wth. We demonstrate that these moment condtons can be partcularly useful when the errors are subject to both weak and strong correlatons, a stuaton that s lkely to arse n practce. The propertes of these GMM estmators have been analysed under d erent crcumstances. Smulated experments have shown that the resultng estmators outperform the conventonal ones, n terms of both bas and RMSE. Ths result s magn ed as the mpact of the factor structure n the total error process ncreases. In addton, larger values of are accompaned by a consderable decrease n bas and RMSE for the estmators put forward n ths paper. Ths s not the case wth the conventonal estmators, the performance of whch s naturally not a ected by the sze of. References [] Ahn, S.C and P. Schmdt, (995) E cent Estmatons of Models for Dynamc Panel Data, Journal of Econometrcs, 68, 5-8. [] Anderson, T.W. and C. Hsao (98) Estmaton of Dynamc Models wth Error Components, Journal of the Amercan Statstcal Assocaton, 76, [3] Arellano, M. (993) On the Testng of Correlated E ects wth Panel Data, Journal of Econometrcs, 59, [4] Arellano, M. and S. Bond (99) Some Tests of Spec caton for Panel Data: Monte Carlo Evdence and an Applcaton to Employment Equatons, Revew of Economc Studes, 58, [5] Arellano, M. and O. Bover (995) Another Look at the Instrumental Varable Estmaton of Error-Component Models, Journal of Econometrcs, 68, 9-5. [6] Ba, J. (005) Panel Data Models wth Interactve Fxed E ects, unpublshed manuscrpt. [7] Baltag, B., Bresson, G. and A. Protte (007) Panel Unt Root Tests and Spatal Dependence, Journal of Appled Econometrcs,,

23 [8] Bover, O. and. Watson (005) Are there economes of scale n the demand for money by rms? Some panel data estmates, Journal of Monetary Economcs, 5, [9] Blundell, R. and S. Bond (998) Intal Condtons and Moment Restrctons n Dynamc Panel Data Models, Journal of Econometrcs, 87, [0] Bun, M.J.G. and J.F. Kvet (006) The E ects of Dynamc Feedbacks on LS and MM Estmator Accuracy n Panel Data Models, Journal of Econometrcs 3, [] Coakley, J., Fuertes, A. and Smth, R. (00) A Prncpal Components Approach to Cross-Secton Dependence n Panels, Unpublshed manuscrpt, Brckbeck College, Unversty of London. [] Conley, T.G. (999) GMM Estmaton wth Cross Sectonal Dependence, Journal of Econometrcs, 9, -45. [3] Goldberger, A.S. (97) Structural Equaton Methods n the Socal Scences, Econometrca, 40, [4] J :: oreskog, K.G. and A.S. Goldberger (975) Estmaton of a model wth multple ndcators and multple causes of a sngle latent varable Journal of the Amercan Statstcal Assocaton, 70, [5] Kapoor, M., Kelejan, H., and I.R. Prucha (007) Panel Data Models wth Spatally Correlated Error Components, Journal of Econometrcs 40, [6] Kelejan, H and D. Robnson (995) Spatal Correlaton: A Suggested Alternatve to the Autoregressve Model, n Anseln, L. and R.J. Florax (ed.), ew Drectons n Spatal Econometrcs, 75-95, Sprnger-Verlag, Berln. [7] Kvet, J.F., (995) On bas, nconsstency, and e cency of varous estmators n dynamc panel data models, Journal of Econometrcs, 68, [8] Lee, L. (007) GMM and SLS Estmaton of Mxed Regressve, Spatal Autoregressve Models, Journal of Econometrcs 37, [9] Moon, H.R. and B. Perron (004) E cent Estmaton of the SUR Contegratng Regresson Model and Testng for Purchasng Power Party, Econometrc Revews, Vol. 3, [0] Pesaran, H. (006) Estmaton and Inference n Large Heterogeneous Panels wth a Multfactor Error Structure, Econometrca, 74, [] Pesaran, H. and E. Tosett (007) Large Panels wth Common Factors and Spatal Correlatons, unpublshed manuscrpt. 3

24 [] Phllps, P. and D. Sul (003) Dynamc Panel Estmaton and Homogenety Testng under Cross Secton Dependence, The Econometrcs Journal, Vol. 6, [3] Phllps, P. and D. Sul (007). Bas n Dynamc Panel Estmaton wth Fxed E ects, Incdental Trends and Cross Secton Dependence, Journal of Econometrcs, 37, [4] Robertson, D. and Symons, J. (000) Factor Resduals n SUR Regressons: Estmatng Panels Allowng for Cross Sectonal Correlaton, unpublshed manuscrpt, Faculty of Economcs and Poltcs, Unversty of Cambrdge. [5] Sara ds, V. Yamagata, T. and D. Robertson (007) A Test of Error Cross Secton Dependence for a Lnear Dynamc Panel Model wth Regressors, unpublshed manuscrpt. [6] Sara ds, V. and D. Robertson (007) On the Impact of Cross Secton Dependence n Short Dynamc Panel Estmaton, unpublshed manuscrpt. [7] Stetzer, F. (98) Specfyng Weghts n Spatal Forecastng Models: The Results of Some Experments, Envronment and Plannng A, 4, [8] Stout, W. (974) Almost Sure Convergence, Academc Press, ew York. [9] Whte, H. (00) Asymptotc Theory for Econometrcans, Academc Press, ew York. [30] Wndmejer, F. (000) E cency Comparsons for a System GMM Estmator n Dynamc Panel Data Models, n R.D.H. Hejmans, D.S.G. Pollock and A. Satora (ed.), Innovatons n Multvarate Statstcal Analyss, Dordrecht: Kluwer Academc Publshers. 4

25 Appendces A Proof of Theorem 3 P The error process s gven by ;t + j w;j j;t + ;t, where E (;t) 0. ote that for E ju j < B u <, we must have jjwjj < Bu <. Hence there are two cases; f the number of non-zero elements n W s nte, such that C O () for all, where C s the number of non-zero elements n row of W, then jw ;jj can be any real number so long as t s su cently bounded. When C grows wth, jw ;jj O for some 0. The correlaton coe cent between t and s +k s gven by t;s Cov ( ;t; +k;s ) 8 >< >: [V ar ( ;t) V ar ( +k;s )] E ( ;t +k;s) E ;t E +;s +k ( P j w ;j w +;j) h P + j w ;j + P + + j w +;j + 0 otherwse, for t s The condton P t;s < s automatcally sats ed when C O () for all, gven that the nonzero values of W are su cently bounded. When C grows wth, the condton jjwjj o mples that jw ;jj o o () and h + P j w ;j + O (), t follows that t;s and therefore jw ;jw +;jj o. + + P j w +;j + o () for su cently large and t s. B Proof of Proposton 8 Assumng that the y ;t process has started a long tme ago, t can be shown that As a result, P j jw;jw+;jj O (). Snce +k where and j (mod ) +. y j;t j + X X s j;t s + s j 0 ;t s (73) s s j 0 j (mod ) + (74) Hence, for DIF GMM we have plm! t3 X plm! t3 y j;t s ;t X j + X j;t s +! X j 0 ;t s ( ;t + j;t) 0, (75) 5

26 usng -5 for s ; :::; T. Furthermore, the covarance between y j;t and y ;t s d erent from zero snce X plm! y j;t sy :t ts+ X! j plm! + X X j;t s + j 0 ;t s ts+! X X ;t + j;t t3 (T s) + 6 0; for s ; :::; T. (76) For SYS GMM we have X plm! y j;t ;t t3! X X X plm! j;t + j 0 ;t ( + ;t + j;t) 0: (77) Furthermore, the covarance between y j;t and y ;t equals X plm! y j;t y ;t t3! X X X plm! j;t + j 0 ;t t3! X X ;t + (T ) j;t (78) C Proof of Proposton 9 De ne o and o. Under Assumptons -4 and we have E [ o o ] o E (o ) 0 snce s non-stochastc. Furthemore, V ar [o o ] E [o o o o0 ] o o0 and Cov o o ; o j o j o o0 j E o j o 0 for 6 j. Hence, from a Weak Law of Large umbers we have X [ o o ] p! 0 Furthermore, followng an approach smlar to (7) we have p p p X [ ] p X X o o ( ) p X o o o [ o ( )] X o p X o + p X ( ) + op () (79) where the last lne follows from the fact that Op, ( ) O p, P o O p () and P o O p (). In the same way we have E o o ;t 6

27 o E o ;t 0, V ar o o ;t o o0 and Cov o o ;t ; o j o j;s o o0 j E o ;t ; j;s o 0 for 6 j and 8 t, P s. As a result, o p! p P o ;t 0 and p P ;t o o ;t + op (). In addton, P o p! p P o ;t 0 and p P ;t o o ;t +op (). Wth these results n mnd, the moment condtons gven n (49) are equal to plm! ts+ X plm! ts+ X y j;t s ;t! j X X X + 0 j f t + j;t s + j 0 ;t s 0 ft + ;t + j;t 0; for s ; :::; T, (80) P snce plm! 0 j 0 and X plm! y j;t s y ;t and ts+ plm! ts+ X! j X X X + 0 j f t s + j;t s + j 0 ;t s! X X X + 0 f t + ;t + j;t t3 (T s) +.(8) For SYS GMM we have X plm! y j;t ;t t3! X X X X plm! 0 j f t + j;t + j 0 ;t + 0 ft + ;t + j;t 0, (8) X plm! t3 plm! 0 t3 y j;t y ;t X! X X X f t + j;t + j 0 ;t 0 j! X X X f t + ;t + j;t (T ) (83) 7

28 SIMULATIO RESULTS Table. Monte Carlo results, b (RMSE) 400 0:5 0:7 0:9 FE IV IV? DIF DIF? SYS SYS? FE IV IV? DIF DIF? SYS SYS? FE IV IV? DIF DIF? SYS SYS? z 3; T 6 :6 :09 :507 :46 :50 :476 :490 :96 :7 :73 :56 :683 :66 :684 :44 7:88 :0 :560 :788 :844 :876 (:364) (5:6) (:75) (:6) (:90) (:59) (:09) (:46) (:4) (:) (:309) (:9) (:64) (:) (:495) (307) (:9) (:557) (:348) (:65) (:) z ; T 6 :53 :497 :5 :344 :49 :457 :487 :85 :588 :697 :453 :659 :635 :679 :409 :0 :058 :430 :76 :8 :869 (:397) (:33) (:300) (:339) (:8) (:9) (:5) (:458) (8:94) (:97) (:450) (:57) (:3) (:9) (:58) (4:9) (3:47) (:693) (:4) (:6) (:7) z 3; T 6 :40 :483 :570 :58 :467 :436 :48 :69 7:43 :739 :34 :66 :6 :669 :389 :34 :736 :35 :649 :797 :858 (:44) (3:34) (:) (:453) (:5) (:76) (:5) (:504) (6) (:00) (:574) (:30) (:77) (:55) (:575) (:0) (7:59) (:757) (:487) (:6) (:5) z 34; T 6 :3 :565 :574 : :404 :43 :475 :58-3:80 :5 :93 :578 :599 :66 :375 :6 :59 :3 :598 :788 :850 (:470) (:84) (:77) (:507) (:76) (:304) (:7) (:533) (5) (5:3) (:6) (:348) (:303) (:75) (:606) (6:57) (5:4) (:785) (:56) (:83) (:7) z 3; T 0 :3 :506 :498 :439 :503 :484 :49 :47 :707 :699 :609 :696 :67 :688 :6 :95 :903 :697 :850 :856 :88 (:3) (:07) (:3) (:45) (:07) (:5) (:07) (:47) (:50) (:9) (:76) (:3) (:5) (:07) (:93) (:55) (:0) (:30) (:6) (:3) (:070) z ; T 0 :303 :58 :50 :385 :494 :47 :489 :459 :73 :70 :53 :680 :653 :694 :608 :3 :93 :59 :84 :835 :878 (:44) (:33) (:60) (:4) (:8) (:65) (:083) (:76) (:663) (:9) (:79) (:30) (:66) (:083) (:38) (5:) (:457) (:49) (:07) (:53) (:08) z 3; T 0 :9 :533 :534 :30 :477 :457 :485 :443 :78 :705 :447 :65 :634 :678 :589 :570 :935 :55 :759 :86 :870 (:83) (:560) (:54) (:306) (:34) (:) (:0) (:35) (:9) (:89) (:374) (:58) (:06) (:0) (:355) (6:4) (3:7) (:50) (:67) (:9) (:097) z 34; T 0 :83 :83 :497 :87 :464 :449 :48 :433 :773 :560 :409 :63 :65 :673 :576 :970 :3 :489 :79 :808 :863 (:307) (0:4) (:58) (:345) (:46) (:35) (:8) (:340) (:43) (4:3) (:44) (:79) (:8) (:8) (:379) (3:8) (5:) (:57) (:305) (:08) (:0) otes: F E s the xed e ects estmator, IV s the Anderson-Hsao estmator and DIF and SY S are the rst-d erenced and system GMM estmators, proposed by Arellano and Bond (99) and Blundell and Bond (998) respectvely. DIF and SY S are estmated n two steps and they use yt and yt 3 as nstruments for yt n the rst-d erenced equatons. Furthermore, SY S uses the optmal weghtng matrx (when 0), as derved n Wndmejer (000). The superscrpt ndcates that the correspondng estmator uses nstrument(s) wth respect to another cross secton. The data generatng process s gven by yt yt + + ft +t +jt, ; ; :::; ; t 48; 47; :::; T wth y; 49 0 and the ntal 50 observatons beng dscarded. d 0; ; t d 0;, ft d 0; f and du [ 0:5; 0:5]. s chosen to ensure that the mpact of the two varance components, and ut, on var (yt) s held constant. f s normalsed to the value of () and s set accordng to (7), such that t changes accordng to the proporton of u attrbuted to the factor structure. alternates between 0:5, 0:7 and 0:9. All experments are based on ; 000 replcatons. 8